# Impossible Mission: Entropy Maximization with Escort Averages

## TL;DR

This paper demonstrates that maximizing deformed entropies with escort averages leads to incorrect thermodynamic relations, suggesting that escort averaging should be avoided in generalized entropy frameworks.

## Contribution

The paper proves that escort averaging in deformed entropy maximization results in incorrect thermodynamic relations, challenging its use in generalized statistical mechanics.

## Key findings

- Escort averaging yields S = ln(Z) instead of S = βU + ln(Z)
- Escort averaging cannot recover standard thermodynamics in the limit
- Using escort averages leads to inconsistent thermodynamic relations

## Abstract

It has recently been a common practice to maximize the deformed entropies through the escort averaging scheme. However, whatever averaging procedure is employed, one should recover the ordinary Shannon maximization results in the appropriate limit of the deformation parameter e.g. $q\to 1$ for the Tsallis and R\'enyi entropies. Otherwise, the very meaning of a consistent generalization becomes at stake. Using only this equivalence, we show that any deformed entropy expression, maximized with the escort averaged constraints, yields that the Shannon entropy $S$ is equal to the logarithm of the ordinary canonical partition function i.e. $S = \ln(Z_S )$ instead of the correct thermodynamic relation $S = \beta U + \ln(Z_S )$. Therefore, we conclude that the use of the escort averaging procedure should be avoided for any deformed entropies, since it cannot even yield the well-known thermodynamic relations of the ordinary canonical formalism.